Negative curves on algebraic surfaces

TL;DR

This paper investigates negative self-intersection curves on algebraic surfaces, showing that over complex numbers such examples are limited, and corrects previous claims of counterexamples to the Bounded Negativity Conjecture.

Contribution

It proves that infinite sequences of negative curves cannot arise from Hecke translates on complex surfaces, refuting earlier counterexamples to the Bounded Negativity Conjecture.

Findings

Negative curves are finitely many on complex surfaces.

Hecke translates do not produce counterexamples over complex numbers.

Previous counterexamples relied on false assumptions about smoothness.

Abstract

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of surfaces for which C^2 is not bounded below are in positive characteristic, and the general expectation is that no examples can arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field. The previous version of this paper claimed to give a counterexample to the Bounded Negativity Conjecture. The idea of the counterexample was to use Hecke translates of a smooth Shimura curve in order to create an infinite sequence of curves violating the Bounded Negativity Conjecture. To this end we applied Hirzebruch Proportionality to all Hecke translates, simultaneously desingularized by a version of Jaffee's Lemma which exists in the literature but which turns out to be false. Indeed, in the new version of the paper, we show that only finitely many Hecke translates of a special subvariety of a Hilbert modular surface remain smooth. This new result is based on work done jointly with Xavier Roulleau, who has been added as an author. The other results in the original posting of this paper remain unchanged.